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The spectral radius of infinite graphs. (English) Zbl 0671.05052
For an infinite graph \(\Gamma\) with vertex set V and finitely bounded valency, the adjacency operator A is well-defined on \(\ell^ 2(V)\) and is bounded and self-adjoint. The spectral radius \(\rho\) (\(\Gamma)\) is the supremum of \(| <x,Ax>|\) over \(\| x\| =1\). Expansion properties of \(\Gamma\) are measured by the isoperimetric constant i(\(\Gamma)\) defined as the infinuum of the ratio of the number of edges having exactly one endpoint in X and \(| X|\) over all finite \(X\subset V\). Some bounds in terms of \(\rho\) (\(\Gamma)\) are derived for i(\(\Gamma)\), thus supporting the idea that in infinite graphs the spectral radius is related to expansion properties of the graph.
Reviewer: D.Cvetković

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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